Acoustic resonator having mode-alignment-cancelled harmonics

ABSTRACT

An acoustic resonator includes a chamber which contains a fluid. The chamber has a geometry which produces self-destructive interference of at least one harmonic in the fluid to avoid shock wave formation at finite acoustic pressure amplitudes. The chamber can have reflective terminations at each end or a reflective termination at only one end. A driver mechanically oscillates the chamber at a frequency of a selected resonant mode of the chamber. The driver may be a moving piston coupled to an open end of the chamber, an electromagnetic shaker or an electromagnetic driver.

BENEFIT OF EARLIER FILING IN THE UNITED STATES

This application is a continuing application of U.S. application Ser.No. 08/226,469 filed Apr. 12, 1994, now abandoned, which is a divisionof U.S. application Ser. No. 07/881,339 filed May 11, 1992, now U.S.Pat. No. 5,319,938, issued Jun. 14, 1994.

REFERENCE TO RELATED APPLICATIONS

This application is related to my copending U.S. application Ser. No.07/665,316 filed Mar. 6, 1991 and my copending U.S. application Ser. No.07/493,580 filed Mar. 14, 1990.

BACKGROUND OF THE INVENTION

1) Field Of Invention

This invention relates to an acoustic resonator in which near-linearmacrosonic waves are generated in a resonant acoustic chamber, havingspecific applications to resonant acoustic compressors.

2) Description of Related Art

My earlier U.S. Pat. No. 5,020,977 is directed to a compressor for acompression evaporation cooling system which employs acoustics forcompression. The compressor is formed by a standing wave compressorincluding a chamber for holding a fluid refrigerant. A travelling waveis established in the fluid refrigerant in the chamber. This travellingwave is converted into a standing wave in the fluid refrigerant in thechamber so that the fluid refrigerant is compressed.

Heretofore, the field of linear acoustics was limited primarily to thedomain of small acoustic pressure amplitudes. When acoustic pressureamplitudes become large, compared to the average fluid pressure,nonlinearities result. Under these conditions a pure sine wave willnormally evolve into a shock wave.

Shock evolution is attributed to a spacial change in sound speed causedby the large variations in pressure, referred to as pressure steepening.During propagation the thermodynamic state of the pressure peak of afinite wave is quite different than its pressure minimum, resulting indifferent sound speeds along the extent of the wave. Consequently, thepressure peaks of the wave can overtake the pressure minimums and ashock wave evolves.

Shock formation can occur for waves propagating in free space, in waveguides, and in acoustic resonators. The following publications focus onshock formation within various types of acoustic resonators.

Temkin developed a method for calculating the pressure amplitude limitin piston-driven cylindrical resonators, due to shock formation (SamuelTemkin, "Propagating and standing sawtooth waves", J. Acoust. Soc. Am.45, 224 (1969)). First he assumes the presence of left and righttraveling shock waves in a resonator, and then finds the increase inentropy caused by the two shock waves. This entropy loss is substitutedinto an energy balance equation which is solved for limiting pressureamplitude as a function of driver displacement. Temkin's theory providedclose agreement with experimentation for both traveling and standingwaves of finite amplitude.

Cruikshank provided a comparison of theory and experiment for finiteamplitude acoustic oscillations in piston-driven cylindrical resonators(D. B. Cruikshank "Experimental investigation of finite-amplitudeacoustic oscillations in a closed tube", J. Acoust. Soc. Am. 52, 1024(1972)). Cruikshank demonstrated close agreement between experimentaland theoretically generated shock waveforms.

Like much of the literature, the work of Temkin and Cruikshank bothassume piston-driven cylindrical resonators of constant cross-sectional(CCS) area, with the termination of the tube being parallel to thepiston face. CCS resonators will have harmonic modes which arecoincident in frequency with the wave's harmonics, thus shock evolutionis unrestricted. Although not stated in their papers, Temkin andCruikshank's implicit assumption of a saw-tooth shock wave in theirsolutions is justified only for CCS resonators.

For resonators with non-harmonic modes, the simple assumption of asawtooth shock wave will no longer apply. This was shown by Weiner whoalso developed a method for approximating the limiting pressureamplitude in resonators, due to shock formation (Stephen Weiner,"Standing sound waves of finite amplitude", J. Acoust. Soc. Am. 40, 240(1966)). Weiner begins by assuming the presence of a shock wave and thencalculates the work done on the fundamental by the harmonics. This workis substituted into an energy balance equation which is solved forlimiting pressure amplitude as a function of driver displacement.

Weiner then goes on to show that attenuation of the even harmonics willresult in a higher pressure amplitude limit for the fundamental. As anexample of a resonator that causes even harmonic attenuation, he refersto a T shaped chamber called a "T burner" used for solid-propellentcombustion research. The T burner acts as a thermally driven 1/2 wavelength resonator with a vent at its center. Each even mode will have apressure antinode at the vent, and thus experiences attenuation in theform of radiated energy through the vent. Weiner offers no suggestions,other than attenuation, for eliminating harmonics. Attenuation is thedissipation of energy, and thus is undesirable for energy efficiency.

Further examples of harmonic attenuation schemes can be found in theliterature of gas-combustion heating. (see for example, Abbott A.Putnam, Combustion-Driven Oscillations in Industry (American ElsevierPublishing Co., 1971)). Other examples can be found in the general fieldof noise control where attenuation-type schemes are also employed, sinceenergy losses are of no importance. One notably different approach isthe work of Oberst, who sought to generate intense sound for calibratingmicrophones (Hermann Oberst, "A method for the production of extremelypowerful standing sound waves in air", Akust. Z. 5.27 (1940)). Oberstfound that the harmonic content of a finite amplitude wave was reducedby a resonator which had non-harmonic resonant modes. His resonator wasformed by connecting two tubes of different diameter, with the smallertube being terminated and the larger tube remaining open. The open endof the resonator was driven by an air jet which was modulated by arotating aperture disk.

With this arrangement, Oberst was able to produce resonant pressureamplitudes up to 0.10 bar for a driving pressure amplitude of 0.02 bar,giving a gain of 5 to the fundamental. The driving waveform, which had a30% error (i.e. deviation from a sinusoid), was transformed to awaveform of only 5% error by the resonator. However, he predicted thatif more acoustic power were applied, then nonlinear distortions wouldbecome clearly evident. In fact, harmonic content is visually noticeablein Oberst's waveforms corresponding to resonant pressure amplitudes ofonly 0.005 bar.

Oberst attributed the behavior of these finite amplitude waves, to thenon-coincidence of the resonator modes and the wave harmonics. Yet, noexplanations were offered as to the exact interaction between theresonator and the wave harmonics. Oberst's position seems to be that thereduced spectral density of the resonant wave is simply the result ofcomparatively little Q-amplification being imparted to the drivingwaveform harmonics. This explanation is only believable for the modestpressure amplitudes obtained by Oberst. Oberst provided no teachings orsuggestions that his methods could produce linear pressure amplitudesabove those which he achieved, and he offered no hope for furtheroptimization. To the contrary,, Oberst stated that nonlinearities woulddominate at higher pressure amplitudes.

A further source of nonlinearity in acoustic resonators is the boundarylayer turbulence which can occur at high acoustic velocities. Merkli andThomann showed experimentally that at finite pressure amplitudes, thereis a critical point at which the oscillating laminar flow will becometurbulent (P. Merkli, H. Thomann, "Transition to turbulence inoscillating pipe flow", J. Fluid Mech., 68, 567 (1975)). Their studieswere also carried out in CCS resonators.

Taken as a whole, the literature of finite resonant acoustics seems topredict that the inherent nonlinearites of fluids will ultimatelydominate any resonant system, independent of the boundary conditionsimposed by a resonator. The literature's prediction of these limits isfar below the actual performance of the present invention.

Therefore, there is a need in the art to efficiently generate very largeshock-free acoustic pressure amplitudes as a means of gas compressionfor vapor-compression heat transfer systems of the type disclosed in myU.S. Pat. No. 5,020,977. Further, many other applications wig the fieldof acoustics, such as thermoacoustic heat engines, can also benefit fromthe generation of high amplitude sinusoidal waveforms.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide acoustic resonatorswhich eliminate shock formation by promoting the destructiveself-interference of the harmonics of a wave, whereby near-linearacoustic pressures of extremely high amplitude can be achieved.

It is another object of the present invention to provide acousticresonators which minimize the nonlinear energy dissipation caused by theboundary layer turbulence of finite acoustic waves.

It is a further object of the present invention to provide acousticresonators which minimize boundary viscous energy dissipation andboundary thermal energy dissipation.

It is a still further object of the present invention to provide anacoustic driving arrangement for achieving high acoustic pressureamplitudes.

It is an even further object of the present invention to provide anacoustic resonator which can maintain near-sinusoidal pressureoscillations while being driven by harmonic-rich waveforms.

The acoustic resonator of the present invention includes a chambercontaining a fluid. The chamber has a geometry which producesdestructive self interference of at least one harmonic in the fluid toavoid shock wave formation at finite acoustic pressure amplitudes. Thechamber has a cross-sectional area which changes along the chamber, andthe changing cross-sectional area is positioned along the chamber toreduce an acoustic velocity of the fluid and/or to reduce boundaryviscous energy dissipation. The

chamber may comprise a resonant chamber for a standing wave compressorused for fluid compression for heat transfer operations.

The acoustic resonator driving system of the invention includes achamber containing a fluid, wherein the chamber has acousticallyreflective terminations at each end. A driver mechanically oscillatesthe chamber at a frequency of a selected resonant mode of the chamber.The acoustic resonator and drive system of the present invention may beconnected to heat exchange apparatus so as to form a heat exchangesystem such as a vapor-compression system.

As described above, the acoustic resonator and acoustic drivingarrangement of the present invention provide a number of advantages andachieve non-linear acoustic pressures of extremely high amplitude. Inparticular, the actual performance of the present invention is farbeyond the results predicted in the literature of finite resonantacoustics.

These and other objects and advantages of the invention will becomeapparent from the accompanying specifications and drawings, wherein likereference numerals refer to like parts throughout.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graphical representation of a resonator having higher modeswhich are harmonics (i.e. integer multiples) of the fundamental;

FIG. 2 is a graphical representation of a resonator having higher modeswhich are not harmonics of the fundamental;

FIG. 3 is a sectional view of an embodiment of a resonator in accordancewith the present invention, which employs an insert as a means of modetuning;

FIG. 4 is a table of measured data for the resonator shown in FIG. 3;

FIG. 5 is a table of theoretical data for the resonator shown in FIG. 3;

FIG. 6 is a sectional view of an embodiment of a resonator in accordancewith the present invention which employs sections of different diameteras a means of mode tuning;

FIG. 7 is a table of measured data for the resonator shown in FIG. 6;

FIG. 8 is a table of theoretical data for the resonator shown in FIG. 6;

FIG. 9 is a sectional view of an embodiment of a resonator in accordancewith the present invention showing further optimizations in resonatorgeometry;

FIG. 10 is a table of theoretical data for the resonator shown in FIG.9;

FIG. 11 is a sectional view of an apparatus used in a resonator drivingsystem in accordance with the present invention, in which the entireresonator is oscillated along its longitudinal axis;

FIG. 12 is a sectional view of the resonator shown in FIG. 9 whichemploys porous materials for enhanced cancellation of higher harmonics;and

FIG. 13 is a sectional view of the resonator and driving system of FIG.11 as connected to heat exchange apparatus to form a heat exchangesystem.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Shock Elimination via Mode-Alignment-Canceled Harmonics

It is well known that "pressure steepening" at high acoustic pressureamplitudes leads to the classic sawtooth waveform of a shock wave. It isalso understood that a sawtooth waveform implies, from Fourier analysis,the presence of harmonics.

If finite amplitude acoustic waves are generated in a constantcross-sectional (CCS) resonator, a shock wave will appear having theharmonic amplitudes predicted by the Fourier analysis of a sawtoothwaveform. At first this would not seem surprising, but it must beunderstood that a CCS resonator has modes which are harmonic (i.e.integer multiples of the fundamental) and which are coincident infrequency with the harmonics of the fundamental. CCS resonators can beconsidered as a special case of a more general class of resonators whosemodes are non-harmonic. Non-harmonic resonators hold a previouslyunharnessed potential for providing extremely high amplitude linearwaves. This potential is realized by non-harmonic resonators which aredesigned to promote the self-destructive interference of the harmonicsof the fundamental.

The present invention employs this principle and provides a newresonator design criterion; to optimize the self-cancellation of waveharmonics. This new design criterion for mode-alignment-canceledharmonics (MACH) eliminates shock formation. MACH resonators haveachieved pressure amplitudes of 100 psi peak-to-peak, with meanpressures of 80 psia, without shock formation. This translates into apeak acoustic pressure amplitude which is 62% of the mean pressure.

Once the MACH design criterion is understood, many different resonatorgeometries can be employed for aligning a resonator's higher modes topromote self-cancellation of harmonics. A straightforward approach forexploiting the MACH principle is to align resonator modes to fallbetween their corresponding harmonics.

The bar graph of FIG. 1 illustrates the relationship between theharmonics of the fundamental and the resonator modes for a CCS 1/2 wavelength resonator. The vertical axis marks the wave harmonics of thewave, and the bar height gives the resonant frequency of the mode. At afundamental frequency of 100 Hz the wave will have harmonics at 200 Hz,300 Hz, 400 Hz, etc. From FIG. 1 it can be seen that the harmonics ofthe wave are coincident in frequency with the modes of the resonator.Stated differently, the nth harmonic of the wave is coincident with thenth mode of the resonator. Consequently, little or no self-destructiveinterference of the wave harmonics will occur, and a shock wave canevolve without restriction. For a well developed shock wave, thepressure amplitude of the 2nd harmonic will be within 6 dB of thefundamental's amplitude.

The bar graph of FIG. 2 illustrates one of many possible arrangementsfor promoting the destructive self-interference of harmonics. In FIG. 2,the resonator modes are aligned to fall between the wave harmonics. Forthis example, the resonator modes have been shifted down in frequency sothat the nth mode lies between harmonics n and n-1. With thisarrangement a large degree of destructive self-interference of the waveharmonics can occur.

FIG. 3 is a sectional view of a resonator which was constructed andtested, and whose modes are shifted down in frequency. The resonator inFIG. 3 is formed by a hollow cylindrical chamber 2, an end flange 4, anend flange 6, and tapered rod insert 8, with all parts being aluminum.Tapered rod insert 8 was welded to end flange 4 with end flange 4 beingwelded to chamber 2. End flange 6 was welded to chamber 2, and wasdrilled to accommodate a process tube and a pressure transducer. Chamber2 has an inside diameter of 5.71 cm, and an inside length of 27 cm.Tapered rod insert 8 has a half-angle end taper of 34.98°, and a lengthof 10 cm, measured from end flange 4. Sharp edges on tapered rod insert8 were rounded off to an arbitrary curvature to reduce turbulence.

Tapered rod insert 8 serves to create a smaller cross-sectional areaalong its length inside of chamber 2. In this way, the resonator of FIG.3 is divided into two sections of different cross-sectional area, eachsection having its own acoustical impedance. This impedance changeresults in a shifting of the resonator modes to non-harmonicfrequencies. The degree to which the modes are shifted can be controlledby varying the diameter and length of tapered rod insert 8. The mannerin which the resonator is driven is described below.

FIG. 4 is a table of measured data obtained for the resonator of FIG. 3.The last column provides a relative measure of the degree of mode shift,by calculating the difference between the frequency "f_(n) " of the nthmode and n times the fundamental frequency "nf₁." The ideal mode shift,for placing the resonator modes at the midpoints between neighboringharmonics, is equal to 1/2 the fundamental frequency. For the FIG. 3resonator, the ideal shift is f₁ /2=166.97 Hz. For CCS resonators, themode shift f_(n) -nf₁ =0 for each mode by definition.

The resonator design of FIG. 3 does not provide ideal mode shifts, butcomes close enough to provide significant results. This is due to thefact that the Fourier sum of the first few harmonics contributes heavilyto shock formation. Thus, significant cancellation of the 2nd, 3rd, and4th harmonics will reduce shock formation greatly. When the resonator ofFIG. 3 was pressurized to 80 psia with gaseous refrigerant HFC-134a,11.8 Watts of acoustic input power was required to achieve a 42 psiapeak-to-peak pressure amplitude (measured at end flange 4). This iswithin 30% of the required driving power predicted by a strictly lineartheory which accounts for only thermal and viscous boundary layerlosses. At these operating conditions the amplitude of the 2nd harmonicwas 20 dB down from the fundamental, with higher harmonics being down 30dB or more.

FIG. 5 is a table of theoretical data which was generated for the FIG. 3resonator. Ideally, f_(n) -nf₁ should be approximately equal to theideal shift for each of the resonator modes. However, it can be seen inFIG. 5 that the degree of mode shifting increases with mode number. Atthe 6th mode, shifting has increased so much that the mode frequency isnow nearly coincident with the 5th harmonic of the wave. With moreadvanced resonator designs, many modes can be simultaneously tuned tolie between the wave harmonics. As the number of properly tuned modesincreases, the resonator's linearity increases.

FIG. 6 is a sectional view of another resonator which was constructedand tested. The resonator in FIG. 6 has a chamber which is formed by asmall diameter section 10, a conical section 12, a large diametersection 14, a conical taper 16, and an end flange 18. The chambercomprising the small diameter section 10, the conical section 12, thelarge diameter section 14, and the conical taper 16 were all machinedfrom a single piece of aluminum. Aluminum end flange 18 was welded toconical end taper 16. Small diameter section 10 has a length of 7.28 cmand a diameter of 3.81 cm. Conical section 12 has a half-angle of 25.63°and an inside length of 3.72 cm. Large diameter section 14 has an insidelength of 13.16 cm and an inside diameter of 7.38 cm. Conical taper 16has a half-angle of 26.08° and an inside length of 2.84 cm. Section 10and section 14 divide the resonator into two sections of differentcross-sectional area, each section having its own acoustical impedance.This design results in a downward shifting of the resonator modes tonon-harmonic frequencies.

The FIG. 6 resonator eliminates the tapered rod insert of FIG. 3,thereby reducing the internal surface area of the resonator, which inturn reduces the thermal and viscous boundary layer losses. The degreeto which the modes are shifted can be controlled by varying thedimensions of section 10, section 14, conical section 12, and taper 16.Taper 16 compensates for excessive downward shifting of the highermodes, by shifting primarily the higher modes up in frequency. Themanner in which the resonator is driven is described below.

FIG. 7 and FIG. 8 are tables of the measured data and theoretical data,respectively, for the resonator of FIG. 6. In comparison with the FIG. 3resonator, the FIG. 6 resonator has improved the tuning of the 2nd, 3rd,and 4th modes, as well as reduced the excessive shifting of highermodes. The FIG. 6 resonator brings the 2nd, 3rd, and 4th modes muchcloser to the ideal shift, and results in improved performance.

When the resonator of FIG. 6 was pressurized to 80 psia, with gaseousrefrigerant HFC-134a, pressure amplitudes of up to 100 psi peak-to-peak(measured at an end 10a of small diameter section 10) were achievedwithout shock formation. However, turbulence was evident, indicatingthat the acoustic velocity was high enough to cause non-laminar flow. Asshown below, resonator geometry can be altered to greatly reduceacoustic velocity. At 60 psi peak-to-peak (measured at the end 10a ofsmall diameter section 10) all harmonics were more than 25 dB down fromthe amplitude of the fundamental, for the FIG. 6 resonator.

In general, the modes of a given resonator geometry can be calculatedfrom the general solution of the wave equation written for both pressureand velocity:

    P(x)=A cos(kx)+B sin(kx)

    V(x)=i/(ρc)(A cos(kx)+B sin(kx))

where i=(-1)^(1/2), ρ=average fluid density, c=speed of sound. Thearbitrary complex constants A and B are found by applying the boundaryconditions of the resonator to the above equations for P(x) and V(x).Resonators embodying the present invention were designed by iteratingP(x) and V(x) in the frequency domain across finite elements of theresonator, until zero velocity is reached at the resonator's end. Asdemonstrated above, the mid-harmonic placement of resonator modesprovides one of many ways to exploit the MACH principle. For more exactpredictions of harmonic cancellation, the harmonics can be treated aswaves traveling within the boundaries of the resonator, while accountingfor their self-interference. The goal of which is to show harmonicself-cancellation as a function of changes in the resonator geometry.

Importance of the MACH Principle

It is revealing to compare the performance of MACH resonators with thatof CCS resonators which do not restrict shock formation. As acomparison, consider the normal evolution to shock formation whichoccurs as a finite amplitude wave propagates. Using the method ofPierce, it is possible to calculate the distance a 60 psi peak-to-peakpressure wave must travel for a fully developed shock wave to evolve(Allan D. Pierce, Acoustics, p.571 (Acoustical Society of America1989)). For a mean pressure of 80 psia (in gaseous HFC-134a), thewaveform will evolve from a sinusoid to a shock after traveling only 22cm, which is less than one traverse of the 27 cm length of the FIG. 6resonator! From this it is easy to appreciate the longstandingassumption that at extremely high amplitudes, intrinsic nonlinearitiesof a gas will dominate any resonator design considerations.

Other Resonator Design Parameters

To efficiently create high amplitude resonant acoustic waves, it isimportant to keep the resonator boundary layer viscous and thermallosses as low as possible. Also, the acoustic velocity, associated witha desired pressure amplitude should be minimized to avoid excessiveturbulence.

For a pure sinusoidal standing wave in a resonator of constantcross-sectional area, the peak acoustic velocity is equal to P/(ρc),where P.tbd.peak acoustic pressure amplitude, ρ.tbd.average fluiddensity, and c.tbd.speed of sound at the average pressure. In practice,the peak acoustic velocity can be decreased by the proper resonatorgeometry. For example, the resonator of FIG. 6 has a peak acousticvelocity equal to 0.82(P/(ρc)) (P being measured at the end 10a of smalldiameter section 10), due to the expansion at the center of the chamberprovided by conical section 12. This increase in cross-sectional areaoccurs just before the velocity maxima at the center of the chamber,thereby lowering the acoustic velocity.

Expansions, like those of the FIG. 6 resonator, have other advantages aswell. When the acoustic velocity is reduced, boundary layer viscouslosses are reduced. Also, the expansion reduces the peak acousticpressure amplitude at end flange 18, thereby reducing boundary layerthermal losses at this end of the resonator. Similarly, the expansionprovided by end taper 16 of FIG. 6 further reduces the boundary layerthermal losses. When the position of an expansion, like conical section12 of FIG. 6, is varied along the length of the resonator, the boundarylayer thermal losses and the boundary layer viscous losses will vary. Ithas been found theoretically that the sum of these losses reaches aminimum when the expansion is centered at approximately 0.3 of thelength of the resonator.

In general, practical energy efficient resonator designs require acompromise between mode tuning for harmonic cancellation, minimizingacoustic velocity, and minimizing thermal and viscous losses. FIG. 9 isa sectional view of a resonator which represents one of a vast number ofpossible compromises between these design parameters.

The FIG. 9 resonator chamber has a conical expansion section 20, acurved expansion section 22, a curved end taper section 24, and an endflange 28. Ports 21a, 21b, such as an inlet and outlet or valves, areprovided at an end 20a of the resonator. Although not shown, such portsare also provided in the resonators of FIGS. 3 and 6. The resonatorchamber is preferably formed by a low thermal conductivity material suchas fiberglass, since this will reduce the boundary layer thermal losses.However, any material, such as aluminum, which can be formed into adesired configuration can be used. The FIG. 9 resonator is similar inprinciple to the FIG. 6 resonator in its method of modal tuning, exceptfor the curved sections which provide greater mode tuning selectivity.This selectivity is due to the varying rate of change of cross-sectionalarea provided by the curved sections, which is explained as follows. Themagnitude of frequency shift of a mode, caused by a given area change,depends on which part of the standing wave pattern encounters the areachange. Each of the many superimposed standing wave patterns in aresonator will encounter a fixed area change at a different point alongits wave pattern. Thus, an area change which tunes one mode properly maycause unfavorable tuning for another mode. Curved sections can providecompensation for this unfavorable tuning by exposing different modes todifferent rates of area change. The term "curved section" is notintended to refer to a specific mathematical surface. Rather, the term"curved section" is understood to mean in general any section whichprovides a rate of change of area, as a function of the longitudinaldimension, whose derivative is non-zero. Any number of mathematicalsurfaces can be employed. It is contemplated that one possible set ofequations for the curved expansion section 22 and curved end taperedsection 24 could be as follows.

In FIG. 9 the constant diameter section at end 20a of the resonator hasan inner diameter of 2.54 cm and is 4.86 cm long. Conical expansionsection 20 is 4.1 cm long and has a 5.8° half-angle. Curved expansionsection 22 is 3.68 cm long. To the right of curved section 22, thediameter remains constant at 5.77 cm over a distance of 11.34 cm. Curvedend taper 24 is 2.16 cm long. To the right of curved end taper 24, thediameter remains constant at 13 cm over a distance of 0.86 cm. Curvedexpansion section 22 was described in a finite element program by theequation D_(n) =D_(n-1) +0.00003(7+n), and curved end taper 24 wasdescribed by the equation D_(n) =D_(n-1) +0.00038(n), where D_(n).tbd.the diameter of the current element, and D_(n-1) .tbd.the diameterof the previous element, and with each element having a length 0.00108meters.

FIG. 10 is a table of theoretical data for the FIG. 9 resonator, whichshows that the point at which modes and harmonics overlap in frequencyhas been significantly extended to higher frequencies.

The FIG. 9 resonator also reduces the acoustic velocity to a value of0.58(P/(ρc)) (P being measured at a small diameter end 20a of theresonator), which represents a significant reduction in acousticvelocity for the desired pressure amplitude. In addition, the FIG. 9resonator reduces the total thermal and viscous energy dissipation ofthe FIG. 6 resonator by a factor of 1.50. Neglecting turbulent losses,the total rate of thermal and viscous energy loss, at a given pressureamplitude, is equal to the acoustic input power required to sustain thatpressure amplitude. Thus, reducing thermal and viscous energy losseswill increase energy efficiency.

Half-Peak Entire-Resonator Driving

The odd modes of a resonator can be effectively driven by mechanicallyoscillating the entire resonator along its longitudinal axis. This isthe preferred method used by the resonators of the present invention.Although the resonators of FIG. 3, FIG. 6, and FIG. 9 could be driven bycoupling a moving piston to an open-ended resonator, this approach hascertain disadvantages which are avoided by the entire resonator drivingmethod.

Entire resonator driving can be understood as follows. If the entireresonator is oscillated along its longitudinal axis, then the end capswill act as pistons. The odd mode pressure oscillations at the twoopposite ends of a double-terminated resonator will be 180° out of phasewith each other. Consequently, when the entire resonator is oscillated,its end caps, or terminations, can be used to drive an odd mode in theproper phase at each end of the resonator. In this way, the fundamentalmode can be effectively driven.

FIG. 11 is a sectional view of one of many approaches which can be usedto drive an entire resonator. In FIG. 11 an electrodynamic shaker ordriver 29 is provided, having a current conducting coil 26 rigidlyattached to end flange 28 of resonator 34, and occupying air gap 30 ofmagnet 32. Magnet 32 is attached to end flange 28 by a flexible bellows36. Bellows 36 maintains proper alignment of coil 26 within air gap 30.

When coil 26 is energized by an oscillating current, the resultingelectromagnetic forces will cause resonator 34 to be mechanicallyoscillated along its longitudinal axis. Magnet 32 can be rigidlyrestrained so as to have infinite mass relative to resonator 34. In thepreferred embodiment, magnet 32 is left unrestrained and thus free tomove in opposition to resonator 34. In either case, an appropriatespring constant can be chosen for bellows 36 to produce a mechanicalresonance equal to the acoustic resonance, resulting in higherelectro-acoustic efficiency. Bellows 36 could be replaced by othercomponents such as flexible diaphragms, magnetic springs, or moreconventional springs made of appropriate materials.

Entire resonator driving reduces the mechanical displacement required toachieve a given pressure amplitude. When driving the entire resonator,both ends of the resonator act as pistons. In most cases, entireresonator driving requires roughly half the peak mechanical displacementwhich would be needed for a single coupled-piston arrangement.

Half-Peak Entire-Resonator (HPER) driving provides the followingadvantages. As discussed above, the proper tuning of modes of a chamberis critical to efficiently achieving high acoustic pressure amplitudes.It follows that this tuning must remain constant during operation.Resonators which are terminated on both ends will maintain precisetuning during operation and throughout the Lifetime of the resonator.

A further advantage relates to the use of HPER driving for acousticcompressors. Since HPER driven chambers are sealed, there are nooil-dependant moving parts that come in contact with the fluid beingcompressed; resulting in an inherently off-free compressor. The suctionand discharge valves needed for acoustic compressors would typically beplaced at the narrow end of a resonator, where the pressure amplitudesare the greatest. For example, valve placement for the resonator of FIG.9 would be positioned at ports 21a, 21b at end 20a. The ratio ofpressure amplitudes at the two ends of the FIG. 9 resonator isapproximately 3:1 (left to right).

Non-Sinusoidal Driving

As discussed above, a properly designed MACH chamber will cause thehigher harmonics of its fundamental to be self canceling. For the samereason, a MACH chamber will tend to cancel out harmonics which may bepresent in the driver's displacement waveform. Thus, MACH chambers canconvert a non-sinusoidal driving displacement into a sinusoidal pressureoscillation. In addition, any mechanical resonance present in a driver,like the driver of FIG. 11, would tend to convert a non-sinusoidaldriving current into a sinusoidal displacement waveform.

In some applications, the use of non-sinusoidal driving signals canresult in greater overall efficiency. For example, the power amplifiersneeded for driving linear motors can be designed to operate veryefficiently in a pulsed output mode. Current pulses can be timed tooccur once every acoustic cycle or to skip several acoustic cycles.

Another type of non-sinusoidal driving, which MACH chambers canfacilitate, is a fluid's direct absorption of electromagnetic energy, asdisclosed in U.S. Pat. No. 5,020,977, the entire content of which ishereby incorporated by reference. Pulsed microwave and infrared energy,when passed through an absorptive fluid, will create acoustic waves inthe fluid. This electromagnetic-to-acoustic conversion will tend toresult in very harmonic-rich acoustic waves. MACH chambers will tend tocancel the resulting harmonics, thereby promoting a sinusoidal pressureoscillation. Electromagnetic pulses can be timed to occur once peracoustic cycle, or to skip several acoustic cycles.

Porous Materials

Porous materials, such as sintered metals, ceramics, and wire meshscreens are commonly used in the field of noise control. Porousmaterials can provide acoustic transmission and refection coefficientswhich vary as a function of frequency and acoustic velocity. Properlyplaced within a resonator, these materials can be used as an aid to modetuning.

FIG. 12 is a sectional view of a resonator 34 illustrating one of manypossible uses of porous materials. In FIG. 12 a porous material 38 isrigidly mounted near end flange 28 of resonator 34. Porous material 38will have a minimal effect on the fundamental of the resonator, whoseacoustic velocity, becomes small near the surface of end flange 28. Thehigher modes of the resonator can have velocity maxima near the positionof porous material 38. Thus, the higher harmonics of the wave canexperience larger reflection coefficients at the porous material and bereflected so as to promote destructive self-interference. Tuning can beadjusted by varying the position of porous material 38 along the lengthof resonator 34.

In this way, a porous material can be used as an aid in optimizing thedestructive self-interference of harmonics. The design flexibilityprovided by porous materials allows more aggressive optimization ofspecific resonator parameters, such as reducing the fundamental'sacoustic velocity, without losing the desired mode tuning.

For microwave driven resonators, porous material 38 could also acttogether with end flange 28 to form a microwave cavity for theintroduction of microwave energy into resonator 34. FIG. 12 illustratesan electromagnetic driver 39 coupled to the resonator 34 by a coaxialcable 41 having a loop termination 41a inside the resonator 34 in thearea between the porous material 38 and end flange 28. The microwaveenergy would be restricted to the area between porous material 38 andend flange 28.

FIG. 13 is a sectional view of resonator 34 and drive apparatus 29 asused in a heat exchange system. In this case, ports 34a and 34b ofresonator 34 are connected to a heat exchange apparatus 45 via conduits47 and 49. Port 34a is provided with a discharge valve 52, and port 34bis provided with a suction valve 54. Discharge valve 52 and suctionvalve 54 will convert the oscillating pressure within resonator 34, intoa net fluid flow through heat exchange apparatus 45. The heat exchangeapparatus may include, for example, a conventional condenser andevaporator, so that the heat exchange system of FIG. 13 may form avapor-compression system.

While the above description contains many specifications, these shouldnot be construed as limitations on the scope of the invention, butrather as an exemplification of one preferred embodiment thereof. Thispreferred embodiment is based on my recognition that acoustic resonatorscan provide significant self-cancellation of harmonics, therebyproviding extremely high amplitude acoustic waves without shockformation. The invention is also based on my recognition that othernonlinearities associated with finite amplitude waves, such asturbulence and boundary layer losses, can be reduced by proper resonatordesign.

Application of the MACH principle can provide nearly completecancellation of wave harmonics. However, the present invention is notlimited to resonators which provide complete cancellation. As shown inthe above specifications, cancellation of a harmonic need not becomplete to obtain shock-free high amplitude acoustic waves. Nor do allharmonics need to be canceled. There is a continuous range of partialharmonic cancellation which can be practiced. Harmonics can be presentwithout shock formation, as long as their amplitudes are sufficientlysmall. Resonators which cancel one, two, or many harmonics could all beconsidered satisfactory, depending on the requirements of a particularapplication. Thus, the scope of the invention is not limited to any onespecific resonator design.

There are many ways to exploit the basic features of the presentinvention which will readily occur to those skilled in the art. Forexample, shifting resonator modes to the midpoint between adjacentharmonics is only one of many ways to exploit the MACH principle.Resonator modes can be shifted to any degree as long as adequateself-destructive interference is provided for a given application.

In addition, many different resonator geometries can support standingwaves and can be tuned to exploit the MACH principle. For example, atoroidal resonator can be tuned by using methods similar to theembodiments of the present invention. Although the present specificationdescribes resonators whose modes are shifted down in frequency, similarresonator designs can shift modes up in frequency. For example, if thediameters of section 10 and section 14 in FIG. 6 are exchanged, then theresonator's modes will be shifted up in frequency rather than down.Furthermore, resonators can be designed to operate in resonant modesother than the fundamental, while still exploiting the MACH principle.Still further, the shock suppression provided by MACH resonators willoccur for both liquids and gases.

Also, it is understood that the application of MACH resonators toacoustic compressors is not limited to vapor-compression heat transfersystems, but can be applied to any number of general applications wherefluids must be compressed. For example, there are many industrialapplications where oil-free compressors are required in order to preventcontamination of a fluid. Finally, many different drivers can be usedwith HPER driven resonators. For example, electromagnetic andpiezoceramic drivers can also provide the forces required for entireresonator driving. In short, any driver that mechanically oscillates theentire resonator and provides the required forces can be used.

Accordingly, the scope of the invention should be determined not by theembodiments illustrated, but by the appended claims and theirequivalents.

What is claimed is:
 1. An electrodynamic driver for an acousticresonator of a type having a chamber containing a fluid, said chamberhaving a geometry which produces destructive self-interference of atleast one harmonic in said fluid to avoid shock wave formation at peakacoustic pressure amplitudes greater than 10% of mean pressure, saidelectrodynamic driver comprising:a current conducting coil rigidlycoupled to the acoustic resonator, said current conducting coil beingperiodically excited; a magnet coupled to said current conducting coil;and a resilient device coupling said magnet to the acoustic resonator, aperiodic excitation of said current conducting coil causing a periodicdisplacement of the acoustic resonator.
 2. An electrodynamic driver asset forth in claim 1, wherein said resilient device comprises a springhaving a spring constant which produces a mechanical resonancecorresponding to a selected acoustic resonance of the acousticresonator.
 3. An electrodynamic driver as set forth in claim 1, whereinsaid magnet is free to move in opposition to the acoustic resonator. 4.An electrodynamic driver as set forth in claim 3, wherein said resilientdevice comprises a spring having a spring constant which produces amechanical resonance corresponding to a selected acoustic resonance ofthe acoustic resonator.
 5. An acoustic resonator driver having anacoustic resonator of a type having a chamber containing a fluid, saidchamber having a geometry which produces destructive self-interferenceof at least one harmonic in said fluid to avoid shock wave formation atpeak acoustic pressure amplitudes greater than 10% of mean pressure,said driver comprising:a current conducting coil being rigidly attachedto an acoustic resonator; a magnet which receives the coil; and a springattaching the magnet to the acoustic resonator; wherein periodicexcitation of the coil causes the resonator and magnet to move inperiodic opposition to one another.
 6. An acoustic resonator driver asset forth in claim 5, wherein the spring constant of the spring ischosen to provide a mechanical resonance, between the resonator and themagnet, whose frequency is equal to the driven acoustic resonancefrequency of the resonator.
 7. An acoustic resonator drive as set forthin claim 5, wherein the magnet is rigidly restrained, thereby preventingthe magnet from moving in periodic opposition to the resonator.
 8. Anacoustic resonator driver as set forth in claim 5, wherein the magnet isunrestrained.